Measure Theory Problem: showing f=0 a.e.

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SUMMARY

The discussion focuses on proving that if a function \( f \in L^2[0,1] \) satisfies \( \int_0^1 f(x)x^n = 0 \) for all non-negative integers \( n \), then \( f = 0 \) almost everywhere. The key insight is the density of continuous functions in \( L^2[0,1] \), which allows the approximation of \( f \) by polynomials. A related problem is also presented, involving \( f \in L^1(\mathbb{T}) \) and the same integral condition, suggesting a connection to Lusin's theorem.

PREREQUISITES
  • Understanding of \( L^2 \) and \( L^1 \) spaces
  • Familiarity with polynomial approximation in functional analysis
  • Knowledge of Hölder's inequality
  • Basic concepts of measure theory and almost everywhere (a.e.) properties
NEXT STEPS
  • Study the density of continuous functions in \( L^2[0,1] \)
  • Learn about polynomial approximation techniques in functional analysis
  • Explore Lusin's theorem and its implications in measure theory
  • Investigate the properties of \( L^1 \) spaces and their relation to Fourier series
USEFUL FOR

Mathematicians, graduate students in analysis, and anyone studying measure theory or functional analysis, particularly those interested in properties of \( L^p \) spaces and approximation theorems.

michael.wes
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Homework Statement


Suppose [tex]f\in L^2[0,1][/tex] and [tex]\int_0^1f(x)x^n=0[/tex] for every n=0,1,2... Show that f = 0 almost everywhere.


Homework Equations



My friend hinted that he used the fact that continuous functions are dense in L^2[0,1], but I'm still stuck.

The Attempt at a Solution



I need help to get started.. I have tried various things with Holder's inequality but I am not getting anywhere. Thanks!
 
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Your friends suggestion is pretty good. Continuous functions can be approximated by polynomials. What polynomials p(x) have that property?
 
Ok, I think I know how to do it now. But there is a follow-up question, which seems a lot harder, but "looks" similar:

Suppose [tex]f\in L^1(\mathbb{T})[/tex] and [tex]\int_0^{2\pi}f(x)x^n=0[/tex] for all n=0,1,2,... Show that f is 0 almost everywhere.

The hint is again to use the density of continuous functions, but also to use part of the proof of Lusin's theorem. I know this is pretty specific so I don't know if you can help... but I would appreciate it. Thanks
 

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